Properties

Label 363.4.a.l
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{2} - 3 q^{3} + 19 q^{4} + 9 q^{5} - 9 \beta q^{6} + 14 \beta q^{7} + 33 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{2} - 3 q^{3} + 19 q^{4} + 9 q^{5} - 9 \beta q^{6} + 14 \beta q^{7} + 33 \beta q^{8} + 9 q^{9} + 27 \beta q^{10} - 57 q^{12} - 41 \beta q^{13} + 126 q^{14} - 27 q^{15} + 145 q^{16} - 51 \beta q^{17} + 27 \beta q^{18} + 84 \beta q^{19} + 171 q^{20} - 42 \beta q^{21} + 90 q^{23} - 99 \beta q^{24} - 44 q^{25} - 369 q^{26} - 27 q^{27} + 266 \beta q^{28} + 51 \beta q^{29} - 81 \beta q^{30} - 188 q^{31} + 171 \beta q^{32} - 459 q^{34} + 126 \beta q^{35} + 171 q^{36} + 133 q^{37} + 756 q^{38} + 123 \beta q^{39} + 297 \beta q^{40} - 21 \beta q^{41} - 378 q^{42} + 42 \beta q^{43} + 81 q^{45} + 270 \beta q^{46} + 72 q^{47} - 435 q^{48} + 245 q^{49} - 132 \beta q^{50} + 153 \beta q^{51} - 779 \beta q^{52} - 45 q^{53} - 81 \beta q^{54} + 1386 q^{56} - 252 \beta q^{57} + 459 q^{58} + 378 q^{59} - 513 q^{60} - 360 \beta q^{61} - 564 \beta q^{62} + 126 \beta q^{63} + 379 q^{64} - 369 \beta q^{65} - 386 q^{67} - 969 \beta q^{68} - 270 q^{69} + 1134 q^{70} - 198 q^{71} + 297 \beta q^{72} - 44 \beta q^{73} + 399 \beta q^{74} + 132 q^{75} + 1596 \beta q^{76} + 1107 q^{78} + 88 \beta q^{79} + 1305 q^{80} + 81 q^{81} - 189 q^{82} - 720 \beta q^{83} - 798 \beta q^{84} - 459 \beta q^{85} + 378 q^{86} - 153 \beta q^{87} + 45 q^{89} + 243 \beta q^{90} - 1722 q^{91} + 1710 q^{92} + 564 q^{93} + 216 \beta q^{94} + 756 \beta q^{95} - 513 \beta q^{96} + 89 q^{97} + 735 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9} - 114 q^{12} + 252 q^{14} - 54 q^{15} + 290 q^{16} + 342 q^{20} + 180 q^{23} - 88 q^{25} - 738 q^{26} - 54 q^{27} - 376 q^{31} - 918 q^{34} + 342 q^{36} + 266 q^{37} + 1512 q^{38} - 756 q^{42} + 162 q^{45} + 144 q^{47} - 870 q^{48} + 490 q^{49} - 90 q^{53} + 2772 q^{56} + 918 q^{58} + 756 q^{59} - 1026 q^{60} + 758 q^{64} - 772 q^{67} - 540 q^{69} + 2268 q^{70} - 396 q^{71} + 264 q^{75} + 2214 q^{78} + 2610 q^{80} + 162 q^{81} - 378 q^{82} + 756 q^{86} + 90 q^{89} - 3444 q^{91} + 3420 q^{92} + 1128 q^{93} + 178 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−5.19615 −3.00000 19.0000 9.00000 15.5885 −24.2487 −57.1577 9.00000 −46.7654
1.2 5.19615 −3.00000 19.0000 9.00000 −15.5885 24.2487 57.1577 9.00000 46.7654
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.a.l 2
3.b odd 2 1 1089.4.a.s 2
11.b odd 2 1 inner 363.4.a.l 2
33.d even 2 1 1089.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.l 2 1.a even 1 1 trivial
363.4.a.l 2 11.b odd 2 1 inner
1089.4.a.s 2 3.b odd 2 1
1089.4.a.s 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{2} - 27 \) Copy content Toggle raw display
\( T_{5} - 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 588 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 27 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 588 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5043 \) Copy content Toggle raw display
$17$ \( T^{2} - 7803 \) Copy content Toggle raw display
$19$ \( T^{2} - 21168 \) Copy content Toggle raw display
$23$ \( (T - 90)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 7803 \) Copy content Toggle raw display
$31$ \( (T + 188)^{2} \) Copy content Toggle raw display
$37$ \( (T - 133)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 1323 \) Copy content Toggle raw display
$43$ \( T^{2} - 5292 \) Copy content Toggle raw display
$47$ \( (T - 72)^{2} \) Copy content Toggle raw display
$53$ \( (T + 45)^{2} \) Copy content Toggle raw display
$59$ \( (T - 378)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 388800 \) Copy content Toggle raw display
$67$ \( (T + 386)^{2} \) Copy content Toggle raw display
$71$ \( (T + 198)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5808 \) Copy content Toggle raw display
$79$ \( T^{2} - 23232 \) Copy content Toggle raw display
$83$ \( T^{2} - 1555200 \) Copy content Toggle raw display
$89$ \( (T - 45)^{2} \) Copy content Toggle raw display
$97$ \( (T - 89)^{2} \) Copy content Toggle raw display
show more
show less